Download Poynting Vector and Power Flow in Electromagnetic Fields

Poynting Vector and Power Flow in Electromagnetic Fields:
Electromagnetic waves can transport energy from one point to another point. The electric and
magnetic field intensities asscociated with a travelling electromagnetic wave can be related to the
rate of such energy transfer.
Let us consider Maxwell's Curl Equations:
Using vector identity
the above curl equations we can write
.............................................(6.35)
In simple medium where
and
are constant, we can write
and
Applying Divergence theorem we can write,
...........................(6.36)
The term
represents the rate of change of energy stored in the electric
and magnetic fields and the term
represents the power dissipation within the volume.
Hence right hand side of the equation (6.36) represents the total decrease in power within the
volume under consideration.
The left hand side of equation (6.36) can be written as
where
2
(W/mt ) is called the Poynting vector and it represents the power density vector associated with
the electromagnetic field. The integration of the Poynting vector over any closed surface gives
the net power flowing out of the surface. Equation (6.36) is referred to as Poynting theorem and
it states that the net power flowing out of a given volume is equal to the time rate of decrease in
the energy stored within the volume minus the conduction losses.
Poynting vector for the time harmonic case:
For time harmonic case, the time variation is of the form
, and we have seen that
instantaneous value of a quantity is the real part of the product of a phasor quantity and
when
is used as reference. For example, if we consider the phasor
then we can write the instanteneous field as
.................................(6.37)
when
E0
is
Let us consider two instanteneous quantities A and B such that
..................(6.38)
where A and B are the phasor quantities.
real.
i.e,
Therefore,
..............................(6.39)
Since A and B are periodic with period
denoted by
, the time average value of the product form AB,
can be written as
.....................................(6.40)
Further, considering the phasor quantities A and B, we find that
and
, where * denotes complex conjugate.
..............................................(6.41)
The poynting vector
can be expressed as
...................................(6.42)
If we consider a plane electromagnetic wave propagating in +z direction and has only
component, from (6.42) we can write:
Using (6.41)
........................................(6.43)
where
and
, for the plane wave under consideration.
For a general case, we can write
.....................(6.44)
We can define a complex Poynting vector
and time average of the instantaneous Poynting vector is given by
.